Thermal storage at low temperatures is cheap and easy with molten salts.
Thermal storage at over 2000 Celsius degrees will be extremely difficult, due to the difficulty of preventing heat losses.
The best would be for the hot body to be stored in argon, because in vacuum it would evaporate and heavier inert gases are expensive. The storage vessel would be very expensive in any case, being made from multiple layers with high temperature resistance, an external surface with high reflectance in red and infrared and other layers with low thermal conductivity, so it is hard to imagine that it could have a size large enough to store much energy.
Another obstacle is that the available power is determined by the emitting surface, not by the volume of the hot body, which is another obstacle for scaling to large amounts of stored energy.
Another obstacle to scaling is that when the hot body cools down the conversion efficiency drops extremely quickly (fourth power), which means that it could store energy only e.g. by being heated and cooled between 2400 and 2100 Celsius degrees.
So only a very small fraction of the thermal capacitance of the hot body can be used, many times lower than when the heat stored in that body would be used to power a closed-cycle heat engine.
So no, these devices may have some useful applications, but energy storage is certainly not one of them, because they are much worse than almost any alternative. Even storing compressed air in a pressure vessel is much more practical.